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A Thermodynamic Basis for the Electronic Properties of Molten Semiconductors: the Role of Electronic Entropy

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A thermodynamic basis for the electronic properties of molten semiconductors: the role of electronic entropy The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Rinzler, Charles C., and A. Allanore. “A Thermodynamic Basis for the Electronic Properties of Molten Semiconductors: The Role of Electronic Entropy.” Philosophical Magazine 97, 8 (December 2016): 561–571 © 2016 Informa UK limited, trading as Taylor & Francis group As Published https://doi.org/10.1080/14786435.2016.1269968 Publisher Taylor & Francis Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/114781 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ A Thermodynamic Basis for the Electronic Properties of Molten Semiconductors: The Role of Electronic Entropy Charles C. Rinzler2 , A. Allanore1 1Massachusetts Institute of Technology, Department of Materials Science and Engineering 77 Massachusetts Avenue, Room 13-5066, Cambridge, MA, USA 02139 e-mail address: [email protected] 617-452-2758 2 e-mail address: [email protected] 617-314-1999 1 A Thermodynamic Basis for the Electronic Properties of Molten Semiconductors: The Role of Electronic Entropy The thermodynamic origin of a relation between features of the phase diagrams and the electronic properties of molten semiconductors is provided. Leveraging a quantitative connection between electronic properties and entropy, a criterion is derived to establish whether a system will retain its semiconducting properties in the molten phase. It is shown that electronic entropy is critical to the thermodynamics of molten semiconductor systems, driving key features of phase diagrams including, for example, miscibility gaps. Keywords: entropy; electronic entropy; thermopower; molten semiconductor PACS: 64, 71, 81 1. Introduction The study of the electronic properties of noncrystalline semiconductor systems has been a frontier in condensed matter physics and materials science since the 1960s [1,2]. Many of the tools developed by the solid state condensed matter community to describe and predict the electronic properties of semiconducting systems depend on the presence of long range order and the lack of strong electron correlation to define computationally tractable solutions to the equations of quantum mechanics. Despite the challenges associated with lack of long-range order on predicting the electronic properties of noncrystalline materials, much progress has been made to phenomenologically and qualitatively describe the basis of semiconducting properties of noncrystalline systems (e.g. amorphous solids and the liquid state). However, the advances in the field over the past 50 years do not currently enable the determination of whether the liquid phase (i.e. the molten state) of a material will behave as a semiconductor. A correlation between certain features of the phase diagrams of systems that exhibit semiconductivity in the molten state has been discussed in the literature since the 1960s, and recent efforts to compile thermodynamic and electronic property data for those systems have confirmed the correlation [3]. However, no explanation for this correlation, nor any quantitative 2 basis, has been proposed to enable the use of phase diagrams to predict the electronic properties of molten systems. We herein apply a quantitative theory connecting the electronic and thermodynamic properties of molten systems to substantiate this correlation. Fultz predicted that the study of the electronic entropy of high temperature systems would be valuable for the prediction of high temperature material properties [4]. In our previous work on molten semiconductors [5], we discussed the various contributions to entropy and demonstrated a quantitative connection between the electronic state entropy*, the electronic properties of molten systems, and their total mixing entropy. We herein leverage this connection to show how electronic entropy is at the origin of the qualitative correlation reported in the prior art. 2. Properties of Molten Semiconductors A description of molten semiconductors is available in reference [5]. These systems exhibit semiconducting properties in the molten state. Many sulfides, oxides, tellurides, selenides, and other systems behave as semiconductors in the molten state. These systems have been studied for over 50 years [1,2]. However, there is as yet no model that provides reliable quantitative prediction of the electronic or thermodynamic properties of these systems, nor whether a system will behave as a semiconductor in the molten state. As early as 1969 a correlation between certain features of phase diagrams of systems that exhibit solid state semiconductor compounds and molten semiconductivity was reported in the literature, as illustrated in Figure 1 [6]. * The reader is invited to consult reference 5 for a discussion on electronic entropy. Electronic entropy is comprised of configurational electronic entropy (the entropy associated with localized electrons) and the electronic state entropy (the entropy associated with the size of the accessible state space of electrons in the system). Electronic state entropy represents the contribution of delocalized electrons as they manifest in the density of states near the Fermi level. This entropy proves to be quantitatively related to electronic transport properties as shown in reference 5. 3 It was observed that systems that remain semiconductors (SC) in the molten state (SC to SC transition) tend to exhibit molten-phase miscibility gaps and congruent melting of a solid semiconductor compound. This has held true across dozens of systems that we have analyzed. Explanations for this observation have historically been qualitative in nature [3,6]. We hereafter analyze this observation within the framework of macroscopic thermodynamics and demonstrate that electronic entropy drives the above-mentioned features of the phase diagrams of molten semiconductors. 3. Connection Between Features of Phase Diagrams and Electronic Properties The enthalpies and entropies of fusion of congruent melting compounds for some systems that exhibit semiconductor-to-semiconductor (SC-SC) and semiconductor-to-metal (SC-M) transitions at melting are shown in Figure 2 in a format analogous to that presented by Iida and Guthrie for elements [7]. Compounds melting to a molten semiconductor (SC-SC) exhibit enthalpies of fusion between 9 and 20 kJ mol-1, while those melting as a metallized liquid (SC-M) exhibit enthalpies of fusion between 24 and 60 kJ mol-1. A clear separation of the enthalpies and entropies of fusion of compounds of different classes of material systems indicates the potential for defining a rule analogous to Richard’s rule [4]. The slope of the line fit to the SC-SC data in Figure 2 is 10.0 J mol-1 K-1, the average entropy of fusion for molten semiconductor systems predicted by Richard for metallic systems (9.6 J mol-1 K-1) [8]. Semiconductor compounds that do not change their electronic behavior upon melting exhibit similar entropy of fusion as metallic compounds, as both retain a similar degree of short-range order across melting. In contrast, the slope of the line fit to the SC-M data in Figure 2 is 41.2 J mol-1 K-1. This reflects the dramatic decrease in short-range order upon 4 melting of these semiconducting compounds, which contributes a large configurational entropy of fusion as well as a large electronic entropy of fusion [3,5,6,9,10]. The difference in enthalpies of fusion between systems that maintain semiconductivity and systems that metallize upon melting can be leveraged to predict whether a system will behave as a molten semiconductor. We begin by analyzing the condition of stability for the metallized or semiconductor state of molten systems. The free energy of mixing (∆!!"#) (see [5] for a reminder on mixing terms) determines the phase diagram of a system and is comprised of enthalpy (∆!!"#) and entropy (∆!!"#) terms: ∆!!"# = ∆!!"# − !∆!!"# (1) We consider a binary mixture that has a congruent-melting solid-state semiconductor compound and ask whether the mixture at the composition of the compound will either retain its semiconducting properties (SC-SC transition, superscript SC) or metallize (SC-M, superscript M) upon melting. The system will behave as a molten semiconductor if its free energy in the !" ! molten semiconductor state (∆!!"#) is lower than its free energy in the metallized state (∆!!"#). Consequently, determining whether a system will behave as a semiconductor in the molten state is equivalent to determining the validity of the inequality: !" ! ∆!!!" < ∆!!"# (2) Breaking the free energy into its enthalpic and entropic components leads to: !" !" ! ! ∆!!"# − !∆!!"# < ∆!!"# − !∆!!"# (3) Rearranging, we can derive a conclusion about the necessary magnitude of the entropy of mixing for a system to behave as a molten semiconductor: ∆!!" − ∆!! ∆!!" > ∆!! + !"# !"# (4) !"# !"# ! 5 We can understand each term in Eq. 4 as a change in entropy that occurs upon mixing. !" The left hand term ∆!!"#, the entropy of mixing of the molten semiconductor state, describes the entropy associated with mixing of end members for the hypothetical molten semiconductor state ! of the system. The first term on the right hand side, ∆!!"#, the entropy of mixing of the metallized molten state, describes the entropy associated with mixing of end members for the hypothetical metallized molten state
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